In Fig. REF , we show two cases (corresponding to two different values of the \(L\)  parameter) in which the symmetry energy is calculated perturbatively around \(d=3\)  from \(d=1\)  to \(d=5\) . It is seen that as the dimension \(d\)  decreases, the symmetry energy decreases correspondingly. Of course, the momentum dependence of the symmetry potential \(U_{\mathrm {sym}}\)  missing in our toy model potential, as mentioned and briefly discussed in section  (see the paragraph after formula () and the analysis after (REF )), may introduce important effects on the density dependence of \(E_{\mathrm {sym}}(\rho )\)  when considering a finite \(\epsilon \) .
These effects are left for further studies. As listed in Tab. REF , here we compare the toy model prediction on the \(L\)  parameter with the results from using the Skyrme and MDI potentials. For the Skyrme-type pseudo potential developed in Ref. [1]}, we find that the linear-\(\epsilon \)  correction of (REF ) is about \(-12.7\epsilon \,\mathrm {MeV}\) , \(-13.6\epsilon \,\mathrm {MeV}\) , or \(-14.5\epsilon \,\mathrm {MeV}\)  for the SP6s, SP6m, or the SP6h parameter set, respectively. For the MDI model [2]}, [3]}, the linear-\(\epsilon \)  correction is about \(-24.7\epsilon \,\mathrm {MeV}\) .
Unlike the circumstance for the magnitude of the symmetry energy listed in Tab. REF , except for the ImMDI potential (which strongly depends on the \(y\)  parameter) the sign of the linear-\(\epsilon \)  correction for the \(L\)  parameter is definite, i.e., the symmetry energy in 2D tends to be hardened (i.e., the correction is positive) at higher densities (since \(\epsilon <0\) ).
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